( NONLINEAR OPTICS PHYC/ECE 568) Homework #4, Due Thu Sept. 24

NONLINEAR OPTICS (PHYC/ECE 568)
Fall 2015 - Instructor: M. Sheik-Bahae
University of New Mexico
Homework #4, Due Thu Sept. 24
Problem 1. SHG in KDP:
a. Calculate the type-I phase matching angle for SHG in KDP using 1.06 m output of a Nd:YAG laser.
b. For a beam radius w0=500 m, calculate the aperture length defined as la= w0 / where  is the
Poynting vector walk-off angle. Obtain the aperture length for w0=15 m and discuss the role of additional
limitations that may be imposed due to diffraction of the beam.
Problem 2. SHG Bandwidth:
a. Calculate the bandwidth  associated with a phase-matched SHG process in terms of the group
velocities vg(1) and vg(21). In the low-depletion approximation, this corresponds to the width of the
Sinc2 function which is taken to be (kL)=2 with L denoting the length of the nonlinear crystal.
Hint: Use the first-order term in the Taylor series expansion of k().
b. Discuss how your results in (a) explains the limitation on the SHG-efficiency when ultrashort laser
pulses are used.
Problem 3. Manley-Rowe Relations for Imaginary (2)
Repeat the derivation of Manley-Rowe relations (section 2.5, Boyd, 3rd ed.) but assume a purely imaginary
(2)
(2)
 =𝑖𝜒𝑖 . Write the new relationship for the three interacting waves involving dIj/dz, and corresponding
photon numbers (1/j) dIj/dz (j=1,2,3). Explain the photon number conservation or lack thereof.
Problem 3:
By taking deff=idi (purely imaginary) and assuming overall permutation symmetry
still applies, the coupled amplitude equations in section 2.5 (Boyd, 3rd ed.) can be
re-arranged to show that:
𝑑𝐼1
𝑑𝑥
𝑑𝐼2
𝑑𝑥
𝑑𝐼3
𝑑𝑥
= −8𝜀0 𝑑𝑖 𝜔1 𝑅𝑒{𝐴3 𝐴∗2 𝐴1∗ }
= −8𝜀0 𝑑𝑖 𝜔2 𝑅𝑒{𝐴3 𝐴∗2 𝐴1∗ }
= −8𝜀0 𝑑𝑖 𝜔3 𝑅𝑒{𝐴3 𝐴∗2 𝐴1∗ }
This in turn leads to
𝑑𝐼1
𝑑𝑥
+
𝑑𝐼2
𝑑𝑥
−
𝑑𝐼3
𝑑𝑥
=0
or 𝐼1 + 𝐼2 − 𝐼3 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
and
𝑑𝐼3
𝜔3 𝑑𝑥
=
𝑑𝐼2
𝜔2 𝑑𝑥
=
𝑑𝐼1
𝜔1 𝑑𝑥
This implies simultaneous absorption of photons from each filed. This is indeed
quantum interference of two-photon absorption (1+2=0 at resonance) and one
photon absorption at 3=0.